A tunable filter may be designed as a closed loop analogue circuit where the output load can change over a wide range. The disclosure further relates to adaptive stability compensation for wide tuning range filters. Filter with programmable cut-off frequency fo (between fo_min and fo_max) are implemented with variable capacitances C1, C2 or resistances R1, R2, R3 as shown in FIG. 1 illustrating a multi-feedback (MFB) low pass filter 100 with a closed loop DC coupled operational amplifier (OPAMP) 101.
The MFB filter 100 includes an operational amplifier 101 having a first (non-inverse, +) input 111, a second (inverse, −) input 112, a first (non-inverse, +) output 121, a second (inverse, −) output 122. A first feedback path including capacitance C1 is coupled between output 122 and input 111. A second feedback path including capacitance C1 is coupled between output 121 and input 112. A third feedback path including resistors R2 and R3 is coupled in parallel to the first feedback path between output 122 and input 111. A fourth feedback path including resistors R2 and R3 is coupled in parallel to the second feedback path between output 121 and input 112. The first (non-inverse, +) input voltage VIN+ is coupled via resistor R1 and the resistor R3 of the third feedback path to the first input 111. The second (inverse, −) input voltage VIN− is coupled via resistor R1 and the resistor R3 of the fourth feedback path to the second input 112. An input loop including the resistors R3 of the third and fourth feedback paths and a further capacitance C2/2 is coupled between the inputs 111, 112.
Generally, it is preferred to program capacitances since in this way Q-factor and noise performances are kept constant versus operating frequency. The operational amplifier (OPAMP) 101 used in the filter 100 has to drive those capacitances C1, C2. In case the fo tuning range is very large, the ratio between maximum and minimum capacitances (Cmax/Cmin) is high, therefore the OPAMP 101 has to cope with very different loading conditions.
FIG. 2 illustrates a possible realization of the RC Filter 100 depicted in FIG. 1 together with its OPAMP 101. The OPAMP 101 may include a non-inverse input path between a drive voltage VDD and ground GND including a first (non-inverse) current source MP+, a first (non-inverse) transistor Q1+ and a second current source (non-inverse) Iin+. A control terminal of Q1+ is coupled to the first input 111 of the OPAMP 101. The OPAMP 101 includes an inverse output path between a drive voltage VDD and ground GND including a second (inverse) transistor QF− and a third current source (inverse) Iout−. A control terminal of QF− is coupled to a first (non-inverse) node D+ of the OPAMP which is located between MP+ and Q1+. A first terminal of QF− is coupled to the second output 122 (VOUT−) of the OPAMP 101. A second terminal of QF− is coupled to the drive voltage VDD. The above described components are additionally used in inverse form as described in the following.
The OPAMP 101 further includes an inverse input path between a drive voltage VDD and ground GND including a first (inverse) current source MP−, a first (inverse) transistor Q1− and a second (inverse) current source Iin−. A control terminal of Q1− is coupled to the second input 112 of the OPAMP 101. The OPAMP 101 includes a non-inverse output path between a drive voltage VDD and ground GND including a second (non-inverse) transistor QF+ and a third (non-inverse) current source Iout+. A control terminal of QF+ is coupled to a first (inverse) node D− of the OPAMP 101 which is located between MP− and Q1−. A first terminal of QF+ is coupled to the first output 121 (VOUT+) of the OPAMP 101. A second terminal of QF+ is coupled to the drive voltage VDD.
A capacitance Cs and a resistor Rs are coupled in parallel between the first terminal of Q1+ and the first terminal of Q1−.
Note that the OPAMP 101 can alternatively be realized as a differential OPAMP as depicted in FIG. 2 or alternatively as a non-differential OPAMP. The non-differential OPAMP 101 has only one first current source MP, one first transistor Q1, one second current source Iin, one third current source Iout, one input and one output without the differentiation of non-inverse and inverse components.
For the specific case of a Multiple Feedback Amplifier (MFB), equations that set the operating frequency, gain H(s) and quality factor of the Filter Q (i.e. its shape) are reported in the following.
            H      ⁡              (        s        )              =                            -                      R            2                          /                  R          1                            1        +                  s          /                      (                          Q              ⁢                                                          ⁢                              ω                o                                      )                          +                              (                          s              /                              ω                o                                      )                    2                                ω      o        =          1                                    C            1                    ⁢                      C            2                    ⁢                      R            2                    ⁢                      R            3                                    Q    =                                        C            2                                C            1                              ⁢                                                  R              2                        ⁢                          R              3                                                            R            2                    +                                    R              3                        ⁡                          (                              1                +                                                      R                    2                                                        R                    1                                                              )                                          
In the case that the filter 100 needs to operate at different frequencies, C2 and C1 capacitance are varied in such a way that 1) for a maximum cutoff frequency fo_max, C2 and C1 are set at minimum value; and that 2) for a Minimum cutoff frequency fo_min, C2 and C1 are set at maximum value.
FIG. 3 illustrates the OPAMP 101 of FIG. 2 with its equivalent RC load. In the OPAMP 101 capacitances between first internal nodes D+, D- and drive voltage VDD are referred to as Co. C2 and C1 are seen by OPAMP 101 as an equivalent capacitance CL at the node VOUT (i.e. VOUT−, VOUT+) as illustrated in FIG. 3 and for sake of simplicity CL_max and CL_min are defined as the equivalent maximum and minimum equivalent capacitances at node VOUT (i.e. VOUT−, VOUT+) in the two extreme cases fo_min (ωmin) and fo_max (ωmax).
The filter 100 is a system in closed loop around its OPAMP 101: in order to have the filter insensitive to OPAMP 101 parameters, open loop gain has to be high at cutoff frequency fo. This requires a minimum gain-bandwidth-product (GBW). So minimum required GBW_MIN is set by maximum frequency fo_max.
As far as OPAMP 101 open loop gain is considered, assuming that the primary pole (dominant pole, or 1st pole) is located at the nodes D+, D− (output of 1st stage), the equivalent capacitance CL set the position of secondary poles S+, S− of the OPAMP 101, hence its stability. The fact that C1 and C2 (hence CL) can have very different values during filter operation makes the problem of OPAMP stability a big challenge and imposes a trade-off on the maximum Gain*Bandwidth (GBW) product of the OPAMP 101 and the Filter tuning range.
To summarise, in a closed Loop Filter there is an intrinsic trade-off between fo_max and tuning range as illustrated in FIGS. 4a and 4b. This is due to the fact that: 1) the minimum required GBW=Goamp*ωp1 product is imposed by fo_max (operating frequency) as shown in FIG. 4a. In fact the OPAMP used into the filter needs to have large open loop gain (Gopamp) at fo_max. 2) The maximum allowed GBW=Goamp*ωp1 product is limited by fo_min as shown in FIG. 4b. In fact, the 2nd pole (wp2) is set by Cmax (when Filter works at fo_min) and the stability condition dictates that ωp2/ωc=3 to have 70 degree phase margin, where ωc is the frequency where Goamp=0. This means that the location of the primary pole is set by the case when the Filter operates at fo_min, thus GBW product will be limited also when Filter is operated at fo_max.
This characteristics is illustrated in FIGS. 4a and 4b. FIGS. 5a, 5b are examples of frequency diagrams that correspond to FIGS. 4a, 4b. FIGS. 5c, 5d show frequency diagrams of a tunable filter according to the disclosure which removes the trade-off between gain bandwidth products versus Cmax/Cmin tuning ranges.